DEMONSTRATIO MATHEMATICA
Vol. 49
No 3
2016
Yılmaz Gündüzalp
SEMI-SLANT SUBMERSIONS FROM ALMOST PRODUCT
RIEMANNIAN MANIFOLDS
Communicated by S. Janeczko
Abstract. In this paper, we introduce semi-slant submersions from almost product
Riemannian manifolds onto Riemannian manifolds. We give some examples, investigate
the geometry of foliations which are arisen from the definition of a Riemannian submersion.
We also find necessary and sufficient conditions for a semi-slant submersion to be totally
geodesic.
1. Introduction
Given a C 8 -submersion π from a Riemannian manifold pM, gq onto a Riemannian manifold pB, g 1 q, there are several kinds of submersions according to
the conditions on it: e.g. Riemannian submersion ([5], [11]), slant submersion
([6], [12], [13]), almost Hermitian submersion [16], quaternionic submersion
[7], etc. As we know, Riemannian submersions are related with physics and
have their applications in the Yang–Mills theory ([3], [17]), Kaluza–Klein theory ([2], [8]), semi-invariant submersion([14]), supergravity and superstring
theories ([9], [10]), etc. In [15], the author studied the slant and semi-slant
submanifolds of an almost product Riemannian manifold. Let pM, g, F q
be an almost product Riemannian manifold. A Riemannian submersion
π : pM, g, F q Ñ pN, g 1 q is called a slant submersion if the angle θpXq between F X and the space kerpπ˚ qp is constant for any nonzero X P Tp M and
p P M [6]. We call θpXq a slant angle. The paper is organized as follows. In
Section 2 we recall some notions needed for this paper. In Section 3 we give
definition of semi-slant submersions and provide examples. We also investigate the geometry of leaves of the distributions. Finally, we give necessary
and sufficient conditions for such submersions to be totally geodesic.
2010 Mathematics Subject Classification: 53C15, 53B20, 53C43.
Key words and phrases: almost product Riemannian manifold, Riemannian submersion,
semi-slant submersion.
DOI: 10.1515/dema-2016-0029
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2. Preliminaries
In this section, we define almost product Riemannian manifolds, recall
the notion of Riemannian submersions between Riemannian manifolds and
give a brief review of basic facts of Riemannian submersions.
Let M be a m-dimensional manifold with a tensor F of type p1, 1q such
that
F 2 “ I, pF ‰ Iq.
Then, we say that M is an almost product Riemannian manifold with
almost product structure F. We put
1
1
P1 “ pI ` F q, P2 “ pI ´ F q.
2
2
Then we get
P1 ` P2 “ I,
P12 “ P1 ,
P22 “ P2 ,
P1 P2 “ P2 P1 “ 0,
F “ P1 ´ P2 .
Thus P1 and P2 define two complementary distributions P1 and P2 . We easily
see that the eigenvalues of F are `1 or ´1.
If an almost product manifold M admits a Riemannian metric g such
that
(1)
gpF X, F Y q “ gpX, Y q
for any vector fields X and Y on M, then M is called an almost product
Riemannian manifold, denoted by pM, g, F q.
Denote the Levi–Civita connection on M with respect to g by ∇. Then,
M is called a locally product Riemannian manifold if F is parallel with
respect to ∇, i.e.,
(2)
∇X F “ 0, X P ΓpT M q r18s.
Let pM, gq and pN, g 1 q be two Riemannian manifolds. A surjective C 8 map π : M Ñ N is a C 8 -submersion if it has maximal rank at any point of M.
Putting Vx “ kerπ˚x , for any x P M, we obtain an integrable distribution V,
which is called vertical distribution and corresponds to the foliation of M
determined by the fibres of π. The complementary distribution H of V,
determined by the Riemannian metric g, is called horizontal distribution.
A C 8 -submersion π : M Ñ N between two Riemannian manifolds pM, gq
and pN, g 1 q is called a Riemannian submersion if, at each point x of M, π˚x
preserves the length of the horizontal vectors. A horizontal vector field X on
M is said to be basic if X is π-related to a vector field X 1 on N. It is clear
that every vector field X 1 on N has a unique horizontal lift X to M and X
is basic.
We recall that the sections of V, respectively H, are called the vertical
vector fields, respectively horizontal vector fields. A Riemannian submersion
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π : M Ñ N determines two p1, 2q tensor fields T and A on M, by the
formulas:
(3)
T pE, F q “ TE F “ h∇vE vF ` v∇vE hF
and
(4)
ApE, F q “ AE F “ v∇hE hF ` h∇hE vF
for any E, F P ΓpT M q, where v and h are the vertical and horizontal projections (see [4]). From (3) and (4), one can obtain
ˆ UW;
(5)
∇U W “ TU W ` ∇
(6)
(7)
(8)
∇U X “ TU X ` hp∇U Xq;
∇X U “ vp∇X U q ` AX U ;
∇X Y “ AX Y ` hp∇X Y q,
for any X, Y P Γppkerπ˚ qK q, U, W P Γpkerπ˚ q. Moreover, if X is basic then
(9)
hp∇U Xq “ hp∇X U q “ AX U.
We note that for U, V P Γpkerπ˚ q, TU V coincides with the second fundamental form of the immersion of the fibre submanifolds and for X, Y P
Γppkerπ˚ qK q, AX Y “ 21 vrX, Y s reflecting the complete integrability of the
horizontal distribution H. It is known that A is alternating on the horizontal
distribution: AX Y “ ´AY X, for X, Y P Γppkerπ˚ qK q and T is symmetric
on the vertical distribution: TU V “ TV U, for U, V P Γpkerπ˚ q.
We now recall the following result which will be useful for later.
Lemma 2.1. (see [4], [11]) If π : M Ñ N is a Riemannian submersion and
X, Y basic vector fields on M, π-related to X 1 and Y 1 on N, then we have
the following properties
1. hrX, Y s is a basic vector field and π˚ hrX, Y s “ rX 1 , Y 1 s ˝ π;
2. hp∇X Y q is a basic vector field π-related to p∇1X 1 Y 1 q, where ∇ and ∇1 are
the Levi–Civita connection on M and N ;
3. rE, U s P Γpkerπ˚ q, for any U P Γpkerπ˚ q and for any basic vector field E.
Let pM, gM q and pN, gN q be Riemannian manifolds and π : M Ñ N is a
smooth map. Then the second fundamental form of π is given by
(10)
p∇π˚ qpX, Y q “ ∇π˚ X π˚ Y ´ π˚ p∇X Y q
for X, Y P ΓpT M q, where we denote conveniently by ∇ the Levi–Civita
connections of the metrics gM and gN . Recall that π is said to be harmonic
if tracep∇π˚ q “ 0 and π is called a totally geodesic map if p∇π˚ qpX, Y q “ 0
for X, Y P ΓpT M q [1]. It is known that the second fundamental form is
symmetric.
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3. Semi-slant submersions
In this section, we define semi-slant submersions from an almost product
Riemannian manifold onto a Riemannian manifold, investigate the integrability of distributions and obtain a necessary and sufficient condition for such
submersions to be totally geodesic map.
Definition 3.1. Let pM, g, F q be an almost product Riemannian manifold and pN, g 1 q be a Riemannian manifold. A Riemannian submersion
1
π : pM, g, F q Ñ pN, g q is called a semi-slant submersion if there is a distribution D1 Ă kerπ˚ such that
kerπ˚ “ D1 ‘ D2 ,
F pD1 q “ D1 ,
and the angle θ “ θpXq between F X and the space pD2 qq is constant for
nonzero X P pD2 qq and q P M, where D2 is the orthogonal complement of D1
in kerπ˚ . We call the angle θ a semi-slant angle.
First, we give some examples of semi-slant submersions.
Example 1. Let π be a slant submersion from an almost product Rieman1
nian manifold pM, g, F q onto a Riemannian manifold pN, g q [6]. Then the
map π is a semi-slant submersion with D2 “ kerπ˚ .
Example 2. Let π be a semi-invariant submersion from an almost product
1
Riemannian manifold pM, g, F q onto a Riemannian manifold pN, g q [14].
Then the map π is a semi-slant submersion with the semi-slant angle cos´1 p0q.
Note that given an Euclidean space R2n with coordinates px1 , . . . , x2n q
on R2n , we can naturally choose an almost product structure F on R2n as
follows:
ˆ
˙
ˆ
˙
B
B
B
B
F
“
“
, F
,
Bx2i
Bx2i´1
Bx2i´1
Bx2i
where i “ 1, . . . , n.
Throughout this section, we will use this notation.
Example 3. Define a map π : R6 Ñ R2 by
πpx1 , . . . , x6 q “ px1 sin α ´ x3 cos α, x4 q,
where 0 ă α ă 90. Then the map π is a semi-slant submersion such that
B
B
B
B
B
D1 “
,
and D2 “
, cos α
` sin α
Bx5 Bx6
Bx2
Bx1
Bx3
with the semi-slant angle cos´1 pαq.
Example 4. Define a map π : R8 Ñ R2 by
ˆ
˙
x1 ´ x3
?
πpx1 , . . . , x8 q “
, x4 .
2
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Then the map π is a semi-slant submersion such that
B
B
B
B
B
B
B
,
,
,
and D2 “
,
`
D1 “
Bx5 Bx6 Bx7 Bx8
Bx2 Bx1 Bx3
` ˘
with the semi-slant angle cos´1 ?12 .
Example 5. Define a map π : R8 Ñ R2 by
πpx1 , . . . , x8 q “ px1 cos α ´ x3 sin α, x2 sin β ´ x4 cos βq,
where α and β are constant. Then the map π is a semi-slant submersion such
that
B
B
B
B
D1 “
,
,
,
and
Bx5 Bx6 Bx7 Bx8
B
B
B
B
D2 “ sin α
` cos α
, cos β
` sin β
Bx1
Bx3
Bx2
Bx4
with the semi-slant angle θ with cos θ “ |sinpα ` βq|.
Example 6. Define a map π : R10 Ñ R4 by
˙
ˆ
x5 ´ x7
x4 ´ x6
?
, x9 , ?
, x10 .
πpx1 , . . . , x10 q “
2
2
Then the map π is a semi-slant submersion such that
B
B
B
B
B
B
B
B
,
,
`
,
,
`
D1 “
and D2 “
Bx1 Bx2
Bx3 Bx4 Bx6 Bx8 Bx5 Bx7
` ˘
with the semi-slant angle cos´1 ?12 .
Example 7. Define a map π : R8 Ñ R2 by
˙
ˆ
x5 ` x7 x6 ´ x8
, ?
.
πpx1 , . . . , x8 q “ x1 , x2 , ?
2
2
Then the map π is a semi-slant submersion such that
B
B
B
B
B
B
D1 “
,
and D2 “ ´
`
,
`
Bx3 Bx4
Bx5 Bx7 Bx6 Bx8
with the semi-slant angle cos´1 p0q.
1
Let π : pM, g, F q Ñ pN, g q be a semi-slant submersion. Then there is a
distribution D1 Ă kerπ˚ such that
kerπ˚ “ D1 ‘ D2 ,
F pD1 q “ D1 ,
and the angle θ “ θpXq between F X and the space pD2 qq is constant for
nonzero X P pD2 qq and q P M, where D2 is the orthogonal complement of D1
in kerπ˚ . Then for X P Γpkerπ˚ q, we have
(11)
X “ P X ` QX,
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Y. Gündüzalp
where P X P ΓpD1 q and QX P ΓpD2 q. For X P Γpkerπ˚ q, we get
F X “ φX ` ωX,
(12)
where φX P Γpkerπ˚ q and ωX P Γppkerπ˚ qK q. For Z P Γppkerπ˚ qK q, we
obtain
F Z “ BZ ` CZ,
(13)
where BZ P Γpkerπ˚ q and CZ P Γppkerπ˚ qK q. For U P ΓpT M q, we have
U “ vU ` hU,
where vU P Γpkerπ˚ q and hU P Γppkerπ˚ qK q. Then
pkerπ˚ qK “ ωD2 ‘ µ,
where µ is the orthogonal complement of ωD2 in pkerπ˚ qK and is invariant
under F. Furthermore,
(14)
φD1 “ D1 , ωD1 “ 0, φD2 Ă D2 , Bppkerπ˚ qK q “ D2 ,
(15)
φ2 ` Bω “ I, C 2 ` ωB “ I, ωφ ` Cω “ 0, BC ` φB “ 0.
We define the covariant derivatives of φ and ω as follows
(16)
ˆ X φY ´ φ∇
ˆ XY
p∇X φqY “ ∇
and
(17)
ˆ XY
p∇X ωqY “ h∇X ωY ´ ω ∇
ˆ X Y “ v∇X Y. Then we easily have
for X, Y P Γpkerπ˚ q, where ∇
1
Lemma 3.1. Let pM, g, F q be a locally product manifold and pN, g q be a
1
Riemannian manifold. Let π : pM, g, F q Ñ pN, g q be a semi-slant submersion.
Then we get
(a)
ˆ X φY ` TX ωY “ φ∇
ˆ X Y ` BTX Y
∇
ˆ X Y ` CTX Y
TX φY ` h∇X ωY “ ω ∇
for any X, Y P Γpkerπ˚ q.
(b)
v∇Z BW ` AZ CW “ φAZ W ` Bh∇Z W
AZ BW ` h∇Z CW “ ωAZ W ` Ch∇Z W
for Z, W P Γppkerπ˚ qK q.
(c)
ˆ X BZ ` TX CZ “ φTX Z ` Bh∇X Z
∇
TX BZ ` h∇X CZ “ ωTX Z ` Ch∇X Z
for X P Γpkerπ˚ q and Z P Γppkerπ˚ qK q.
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Theorem 3.1. Let π be a semi-slant submersion from an almost product
1
Riemannian manifold pM, g, F q onto a Riemannian manifold pN, g q. Then
the distribution D1 is integrable if and only if we have
ˆ XY ´ ∇
ˆ Y Xq “ CpTY X ´ TX Y q
ωp∇
for X, Y P ΓpD1 q.
Proof. For X, Y P ΓpD1 q and Z P Γppkerπ˚ qK q, since rX, Y s P Γpkerπ˚ q,
from (5), (12) and (13) we get
gpF rX, Y s, Zq “ gpF ∇X Y ´ F ∇Y X, Zq
ˆ X Y ´ F TY X ´ F ∇
ˆ Y X, Zq
“ gpF TX Y ` F ∇
ˆ X Y ` ω∇
ˆ XY
“ gpBTX Y ` CTX Y ` φ∇
ˆ Y X ´ CTY X ´ ω ∇
ˆ Y X, Zq
´ BTY X ´ φ∇
ˆ X Y ´ CTY X ´ ω ∇
ˆ Y X, Zq.
“ gpCTX Y ` ω ∇
Therefore, we have the result.
Theorem 3.2. Let π be a semi-slant submersion from an almost product
1
Riemannian manifold pM, g, F q onto a Riemannian manifold pN, g q. Then
the slant distribution D2 is integrable if and only if we get
ˆ XY ´ ∇
ˆ Y Xq ` BpTX Y ´ TY Xqq “ 0
P pφp∇
for X, Y P ΓpD2 q.
Proof. For X, Y P ΓpD2 q and Z P ΓpD1 q, since rX, Y s P Γpkerπ˚ q, from (5),
(12) and (13) we obtain
gpF rX, Y s, Zq “ gpF ∇X Y ´ F ∇Y X, Zq
ˆ X Y ´ BTY X ´ φ∇
ˆ Y X, Zq
“ gpBTX Y ` φ∇
which proves assertion.
Lemma 3.2. Let π be a semi-slant submersion from a locally product
1
manifold pM, g, F q onto a Riemannian manifold pN, g q. Then the distriˆ X φY ´ ∇
ˆ Y φXq “ 0 and
bution D1 is integrable if and only if we obtain Qp∇
TX φY “ TY φX for X, Y P ΓpD1 q.
Proof. For X, Y P ΓpD1 q, from (2), (5), (12) and (14) we obtain
F rX, Y s “ ∇X F Y ´ ∇Y F X
ˆ X φY ´ TY φX ´ ∇
ˆ Y φX
“ TX φY ` ∇
which proves assertion.
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Lemma 3.3. Let π be a semi-slant submersion from a locally product mani1
fold pM, g, F q onto a Riemannian manifold pN, g q. Then the slant distribution
D2 is integrable if and only if we have
ˆ X φY ´ ∇
ˆ Y φX ` TX ωY ´ TY ωXq “ 0
P p∇
for X, Y P ΓpD2 q.
Proof. For X, Y P ΓpD2 q and Z P ΓpD1 q, since rX, Y s P Γpkerπ˚ q, from (2),
(5), (6) and (12) we obtain
gpF rX, Y s, Zq “ gp∇X F Y ´ ∇Y F X, Zq
ˆ X φY ´ TY ωX ´ ∇
ˆ Y φX, Zq.
“ gpTX ωY ` ∇
Therefore, the result follows.
Theorem 3.3. Let π be a semi-slant submersion from an almost product
1
Riemannian manifold pM, g, F q onto a Riemannian manifold pN, g q. Then
we get
(18)
φ2 X “ cos2 θX
for X P ΓpD2 q, where θ denotes the semi-slant angle of D2 .
Proof. For X P ΓpD2 q, we can write
(19)
cos θpXq “
}φX}
.
}F X}
By using (12), (19) and (1) we get
(20)
gpφ2 X, Xq “ gpφX, φXq
“ cos2 θpXqgpF X, F Xq
“ cos2 θpXqgpX, Xq
for X P ΓpD2 q. Since g is Riemannian metric, from (20) we have
φ2 X “ cos2 θpXqX,
X P ΓpD2 q.
Corollary 3.1. Let π be a semi-slant submersion from an almost product
1
Riemannian manifold pM, g, F q onto a Riemannian manifold pN, g q. Then
we have
gpφX, φY q “ cos2 θgpX, Y q,
gpωX, ωY q “ sin2 θgpX, Y q
for X, Y P ΓpD2 q.
From (15) and (18) we have
Corollary 3.2. Let π be a semi-slant submersion from an almost product
Riemannian manifold pM, g, F q onto a Riemannian manifold pN, g 1 q. Then
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π is a semi-slant submersion if and only if there exists a constant k P r0, 1s
such that
Bω “ kI.
If π is a semi-slant submersion, then k “ sin2 θ, where θ denotes the semi-slant
angle of D2 .
Theorem 3.4. Let π be a semi-slant submersion from a locally product
1
manifold pM, g, F q onto a Riemannian manifold pN, g q. Then the distribution
kerπ˚ defines a totally geodesic foliation if and only if
ˆ X φY ` TX ωY q ` CpTX φY ` h∇X ωY q “ 0
ωp∇
for X, Y P Γpkerπ˚ q.
Proof. For X, Y P Γpkerπ˚ q, from (2), (5), (6) and (12) we obtain
∇X Y “ F ∇X F Y
ˆ X φY ` TX ωY ` h∇X ωY q.
“ F pTX φY ` ∇
Using (12) and (13), we have
ˆ X φY ` ω ∇
ˆ X φY
∇X Y “ BTX φY ` CTX φY ` φ∇
` φTX ωY ` ωTX ωY ` Bh∇X ωY ` Ch∇X ωY.
Thus, we get
ˆ X φY ` TX ωY q “ 0.
∇X Y P Γpkerπ˚ q ô CpTX φY ` h∇X ωY q ` ωp∇
From (2), (7), (8), (12) and (13) we have
Theorem 3.5. Let π be a semi-slant submersion from a locally product
1
manifold pM, g, F q onto a Riemannian manifold pN, g q. Then the distribution
pkerπ˚ qK defines a totally geodesic foliation if and only if
φpv∇X BY ` AX CY q ` BpAX BY ` h∇X CY q “ 0
for X, Y P Γppkerπ˚ qK q.
In a similar way we have the following.
Theorem 3.6. Let π be a semi-slant submersion from a locally product
1
manifold pM, g, F q onto a Riemannian manifold pN, g q. Then the distribution
ˆ X φY `BTX φY q “ 0
D1 defines a totally geodesic foliation if and only if Qpφ∇
ˆ
and CTX φY ` ω ∇X ωY “ 0, for X, Y P ΓpD1 q.
From Theorem 3.4 we have the following result.
Theorem 3.7. Let π be a semi-slant submersion from a locally product
1
manifold pM, g, F q onto a Riemannian manifold pN, g q. Then the distribution
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D2 defines a totally geodesic foliation if and only if
ˆ X φY ` TX ωY q ` BpTX φY ` h∇X ωY qq,
0 “ P pφp∇
ˆ X φY ` TX ωY q ` CpTX φY ` h∇X ωY q
0 “ ωp∇
for X, Y P ΓpD2 q.
Theorem 3.8. Let π be a semi-slant submersion from a locally product
1
manifold pM, g, F q onto a Riemannian manifold pN, g q. If the tensor ω is
parallel, then we have
TφX φX “ cos2 θTX X
for X P Γpkerπ˚ q.
Proof. For X, Y P Γpkerπ˚ q, from Lemma 3.1(a) we get
TX φY “ CTX Y.
So that interchanging the role of X and Y,
TY φX “ CTY X.
Hence
TX φY “ TY φX.
Substituting Y by φX and using (18),
TφX φX “ cos2 θTx X.
Finally we give necessary and sufficient conditions for a semi-slant submersion to be totally geodesic. Recall that a differentiable map π between
1
Riemannian manifolds pM, gq and pB, g q is called a totall geodesic map if
p∇π˚ qpX, Y q “ 0 for all X, Y P ΓpT M q.
Theorem 3.9. Let π be a semi-slant submersion from a locally product
1
manifold pM, g, F q onto a Riemannian manifold pN, g q. Then π is a totally
geodesic map if and only if
ˆ X φY ` TX ωY q ` CpTX φY ` h∇X ωY q,
0 “ ωp∇
ˆ X BZ ` TX CZq ` CpTX BZ ` h∇X CZq
0 “ ωp∇
for X, Y P Γpkerπ˚ q and Z P Γppkerπ˚ qK ).
Proof. For Z1 , Z2 P Γppkerπ˚ qK q, since π is a Riemannian submersion, from
(10) we obtain
p∇π˚ qpZ1 , Z2 q “ 0.
For X, Y P Γpkerπ˚ q, using (2), (10) and (12) we have
p∇π˚ qpX, Y q “ ´π˚ p∇X Y q “ ´π˚ pF ∇X pφY ` ωY q.
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From (5), (6), (12) and (13) we get
ˆ X φY ` ω ∇
ˆ X φY ` BTX φY ` CTX φY
p∇π˚ qpX, Y q “ ´π˚ pφ∇
` φTX ωY ` ωTX ωY ` Bh∇X ωY ` Ch∇X ωY q.
Thus, we have
ˆ X φY ` TX ωY q ` CpTX φY ` h∇X ωY q “ 0.
p∇π˚ qpX, Y q “ 0 ô ωp∇
For X P Γpkerπ˚ q and Z P Γppkerπ˚ qK q, using again (2), (10) and (12) we
obtain
p∇π˚ qpX, Zq “ p∇π˚ qpZ, Xq “ ´π˚ p∇X Zq “ ´π˚ pF ∇X pBZ ` CZq.
From (5), (6), (12) and (13) we get
ˆ X BZ ` ω ∇
ˆ X BZ ` BTX BZ ` CTX BZ
p∇π˚ qpX, Zq “ ´π˚ pφ∇
` φTX CZ ` ωTX CZ ` Bh∇X CZ ` Ch∇X CZq.
Thus, we obtain
ˆ X BZ ` TX CZq ` CpTX BZ ` h∇X CZq “ 0.
p∇π˚ qpX, Zq “ 0 ô ωp∇
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Y. Gündüzalp
DEPARTMENT OF MATHEMATICS
DICLE UNIVERSITY
21280, DIYARBAKIR, TURKEY
E-mail: [email protected]
Received September 22, 2014.
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